## Extensions of the duality between minimal surfaces and maximal surfaces.(English)Zbl 1211.53010

Author’s abstract: As a generalization of the classical duality between minimal graphs in $$E^{3}$$ and maximal graphs in $$L^{3}$$, we construct the duality between graphs of constant mean curvature $$H$$ in Bianchi-Cartan-Vranceanu space $$E^{3} (\kappa , \tau )$$ and spacelike graphs of constant mean curvature $$\tau$$ in Lorentzian Bianchi-Cartan-Vranceanu space $$L^{3}(\kappa , H)$$.

### MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35J60 Nonlinear elliptic equations 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

### Keywords:

mean curvature; bundle curvature; duality
Full Text:

### References:

 [1] Albujer A.L., Alías J.: Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces. J. Geom. Phys. 59(5), 620–631 (2009) · Zbl 1173.53025 [2] Alías L.J., Dajczer M., Rosenberg H.: The Dirichlet problem for constant mean curvature surfaces in Heisenberg space. Calc. Var. 30, 513–522 (2007) · Zbl 1210.53010 [3] Araújo H., Leite M.L.: How many maximal surfaces do correspond to one minimal surface?. Math. Proc. Camb. Phil. Soc. 146, 165–175 (2009) · Zbl 1165.53007 [4] Alías L.J., Palmer B.: A duality result between the minimal surface equation and the maximal surface equation. An. Acad. Bras. Cienc. 73(2), 161–164 (2001) · Zbl 0999.53007 [5] Abresch U., Rosenberg H.: Generalized Hopf differentials. Mat. Contemp. 28, 1–28 (2005) · Zbl 1118.53036 [6] Bekkar M., Bouziani F., Boukhatem Y., Inoguchi J.: Helicoids and axially symmetric minimal surfaces in 3-dimensional homogeneous spaces. Differ. Geom. Dyn. Syst. 9, 21–39 (2007) · Zbl 1159.53335 [7] Calabi, E.: Examples of Bernstein problems for some non-linear equations. Proc. Sympos. Pure Math. 15, Amer. Math. Soc., Providence, RI, 223–230 (1970) · Zbl 0211.12801 [8] Caddeo R., Piu P., Ratto A.: SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogenous spaces. Manuscripta Math. 87, 1–12 (1995) · Zbl 0827.53009 [9] Choi H.I., Treibergs A.: Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. J. Differ. Geom. 32(3), 775–817 (1990) · Zbl 0717.53038 [10] Daniel, B.: The Gauss map of minimal surfaces in the Heisenberg group. Int. Math. Res. Notices. to appear · Zbl 1209.53048 [11] Daniel B.: Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. 82(1), 87–131 (2007) · Zbl 1123.53029 [12] Fang, Y.: Lectures on minimal surfaces in R 3. In: Proceedings of the Center for Mathematics and Its Applications, The Australian National University, pp. 19–20. (1986) [13] Figueroa C., Mercuri F., Pedrosa R.: Invariant surfaces of the Heisenberg groups. Ann. Mat. Pura Appl. 177(4), 173–194 (1999) · Zbl 0965.53042 [14] Han Z.-C., Tam L.-F., Treibergs A., Wan T.: Harmonic maps from the complex plane into surfaces with nonpositive curvature. Comm. Anal. Geom. 3(1), 85–114 (1995) · Zbl 0843.58028 [15] Lee, H.: Maximal surfaces in Lorentzian Heisenberg space. Differ. Geom. Appl. to appear · Zbl 1216.53055 [16] Lee, H.: Applications of the twin correspondence between CMC H-graphs in E 3({$$\kappa$$}, {$$\tau$$}) and spacelike CMC {$$\tau$$}-graphs in L 3({$$\kappa$$}, H), preprint [17] López F.J., López R., Souam R.: Maximal surfaces of Riemann type in Lorentz-Minkowski space L 3. Michigan Math. J. 47(3), 469–497 (2000) · Zbl 1029.53014 [18] Nitsche C.C.: Elementary proof of Bernstein’s theorem on minimal surfaces. Ann. Math. 66, 543–544 (1957) · Zbl 0079.37702 [19] Osserman R.: A Survey of Minimal Surfaces, Second Edition, pp. 31–37. Dover Publications Inc, New York (1986) [20] Palmer B.: Spacelike constant mean curvature surfaces in pseudo-Riemannian space forms. Ann. Glob. Anal. Geom. 8(3), 217–226 (1990) · Zbl 0711.53048 [21] Shiffman M.: On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes. Ann. Math. 63, 77–90 (1956) · Zbl 0070.16803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.